reserve k, l, m, n, i, j for Nat,
  K, N for non empty Subset of NAT,
  Ke, Ne, Me for Subset of NAT,
  X,Y for set;
reserve f for Function of Segm n,Segm k;
reserve x,y for set;

theorem Th47:
n>=1 implies n block 2 = 1/2 * (2 |^ n - 2 )
proof
  defpred P[Nat] means $1 block 2 = 1/2 * (2 |^ $1-2);
A1: for k be Nat st k>=1 & P[k] holds P[k + 1]
  proof
    let k be Nat such that
A2: k >=1 and
A3: P[k];
    (k+1) block 2 = 2*( k block (1+1)) + (k block 1) by Th46
      .=2 * (1/2 * (2 |^ k - 2)) + 1 by A2,A3,Th32
      .=1/2* (2* 2 |^ k - 2)
      .=1/2* (2 |^ (k+1)-2) by NEWTON:6;
    hence thesis;
  end;
A4: P[1] by Th29;
  for k be Nat st k>=1 holds P[k] from NAT_1:sch 8(A4,A1);
  hence thesis;
end;
